Mathematics Contest Worksheet - Bluffton - 2009 Page 4
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65. The numbers 1, 3, 6, 10, 15, 21, . . . are known as triangular numbers. What is the largest
triangular number less than 300?
Solution 1: This can be found by “brute force,” by simply adding consecutive integers until 300
is reached, at 276.
n(n+1)
=
, so we want n(n +1) < 600.
Solution 2: The formula for the nth triangular number is T
√
n
.
2
= 24.5. Noting that T
= 300, we know that the
That means that n should be close to
600
24
= 276.
answer must be T
23
6. Find the area of a rectangle if its length is 24 and its diagonal has length 25.
Solution: Call the width of the rectangle w. By the Pythagorean theorem, 24
+ w
= 25
2
2
2
, so
w = 7, and the area is (7)(24) = 168 square units.
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7. Among the adult residents of Mathville, half of the men are married to women, and
of the
3
women are married to men. (No one is married to more than one person.) If there are 240 men
in the village, how many total adults live in Mathville?
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Solution: 120 of the 240 men are married to women, so 120 must be
of the women. That means
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there are 360 women, for a total of 240 + 360 = 600 adults.
Section B. Each correct answer is worth 2 points.
8. Quadrilateral MATH is inscribed in the circle as shown. Name the smallest angle in the
quadrilateral and give its measure.
A
◦
Solution: Quadrilateral MATH is called a cyclic quadrilateral,
99
meaning that its vertices lie on a circle. Cyclic quadrilaterals
◦
M
103
have the property that opposite angles are supplementary, so
T
◦
the smallest angle is T , with measure 77
.
H
9. I bought 3 books at $19.90 each. My total cost, including sales tax, was $63.58. What was the
sales tax rate (to the nearest half of one percent)?
Solution: Before tax, the books cost $59.70. If the tax rate is r , then 59.70(1 + r ) = 63.58
.
(after rounding the right side to the nearest penny). Therefore, 1 + r
=
63.58
= 1.06499 . . ., so
59.70
r = 0.065 = 6.5%.
10. Let z = |−x| − |2y| + x y. Evaluate z if
(a) x = 3 and y = −2. (b) x = −3 and y = −4. (For credit, both answers must be correct.)
Solution: (a) With x = 3 and y = −2, z = |−3| − |−4| + (3)(−2) = 3 − 4 − 6 = −7.
(b) With x = −3 and y = −4, z = |3| − |−8| + (−3)(−4) = 3 − 8 + 12 = 7.
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